# [EM] MaxMinPA

Michael Ossipoff email9648742 at gmail.com
Thu Oct 13 08:58:21 PDT 2016

```I looked at the rankings version because it seems to me that the methods
with deluxe properties are ranking methods.

In the standard chicken dilemma example, it seems to me that C wins even if
the A voters & the B voters all cooperate.

...because A's min pairwise approval = A, in the A,B pair.

...& B's min pairwise approval
= B, in the B,A pair.

...& C's pairwise approval = C in both of C's pairings.

Michael Ossipoff
On Oct 12, 2016 2:17 PM, "Forest Simmons" <fsimmons at pcc.edu> wrote:

> The following method is based on score or range style ballots.  I believe
> it satisfies the FBC, Plurality, the CD, Monotonicity, Participation,  Clone
> Independence, and the IPDA.  It reduces to ordinary Approval when only
> the extreme ratings are used for all candidates.
>
>
>
> I call it MinMaxPairwiseApproval or MinMaxPA for short.
>
>
>
> It is based on a concept of “pairwise approval.”
>
>
>
> A zero to 100% cardinal ratings ballot contributes the following amount to
> the “pairwise approval of candidate X relative to candidate Y”:
>
>
>
> The amount is either …
>
> 100% if X is rated strictly above Y, or
>
> Zero if X is rated strictly below Y, or
>
> Their common rating if they are rated equally.
>
>
>
> According to this definition, the ballot’s contribution to the pairwise
> approval of X relative to itself is simply the ballot’s rating of X, since
> it is rated equally with itself.
>
>
>
> The method elects the candidate whose minimum pairwise approval (relative
> to all candidates including self) is maximal.
>
>
>
> The motivation for this idea is the question, “If candidates X and Y were
> the only two candidates with any significant chance of winning the
> election, what is the probability that the ratings ballot voter would want
> X approved (in a Designated Strategy Voting system, say)?”
>
>
>
> If the voter rated X over Y, this probability would be 100 percent.
>
> If the voter rated Y over X, this probability would be zero.
>
> If the voter rated both X and Y at 100 percent, this probability would be
> 100 percent.
>
> If the voter rated them both at zero, she would want neither of the
> approved.
>
> If she rated them both at 50%, then our best guess is that there is a
> fifty-fifty chance that she would approve X.
>
> Etc.
>
>
>
> Whatever nice properties the method has depends solely on its definition,
> not the motivation for the definition, so please explore it with an open
> mind.
>
>
>
> Tomorrow, when I have more time, I’ll give some examples.
>
>
> Enjoy,
>
>
> Forest
>
>
> P.S.
>
>
> The rules can be modified for ranked preference ballots:
>
>
> The amount (per ballot) of approval of X relative to Y  is either ...
>
>
> 100 percent if X is ranked ahead of Y or equal top with Y
>
> zero if Y is ranked ahead of X or equal bottom with X
>
> 50 percent if both are ranked equally and strictly between top and bottom.
>
> Smith//MaxMinPA may be a nice method that trades the FBC and possibly
> other nice properties for the Condorcet Criterion.
>
>
> ----
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>
>
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